Why is $f_\epsilon(u) \in H_0^{1,2}(\Omega)$?

Question

I have to proof that $ u \in H_0^{1,2}(\Omega)$ implies $|u| \in H_0^{1,2}(\Omega)$ . I can define $f_\epsilon(u)=\sqrt{\epsilon^2+u^2}-\epsilon$ .

it follows that $\nabla f_\epsilon(u)=\frac{u}{\sqrt{\epsilon^2+u^2}}\nabla u$.

Answer 1

Assuming you can justify that $f_\epsilon(u)$ is weakly differentiable with that gradient, there should be no issue with proving they are in $L^2$. Note for example $$ (f_\epsilon (x))^2 ≤ (f_\epsilon (x)+\epsilon)^2 = \epsilon^2 + x^2 $$ and for the second, note that $\frac{u}{\sqrt{\epsilon^2+u^2}}\in L^\infty$.